A class of real cocycles having an analytic coboundary modification
Let \((X,{\mathcal B},\mu)\) be the unit interval with Lebesgue measurable sets and Lebesgue measure and \(T: X\to X\) an irrational rotation, \(Tx= x+ \alpha\text{ mod }1\). Denote by \(G\) a locally compact Abelian metric group with Haar measure \(m\) (usually \(G= \mathbb{R}\) or \(S^ 1\)), and \(\phi: X\to G\) a measurable function. Then \(T_ \phi: X\times G\to X\times G\) defined by \[ T_ \phi(x, g)= (Tx, g\cdot \phi(x)) \] is called a \(G\)- extension of \(T\) with cocycle \(\phi\). If \(G= S^ 1\) then \(T_ \phi\) is called an Anzai skew product. Two cocycles \(\phi\) and \(\psi\) are cohomologous if there exists a measurable function \(f: X\to G\) satisfying \(\phi/\psi(x)= f(Tx)/f(x)\), \(x\in X\). In Israel J. Math. 80, No. 1-2, 33-64 (1992; Zbl 0774.28008), the same authors constructed two Anzai skew products that are weakly isomorphic but not isomorphic (in fact a class of such examples over a dense \(G_ \delta\) of admissible base rotations). It was shown that the cocycles in those examples may be smoothed to give \(C^ \infty\) cocycles with the same properties. In this paper the authors abstract these properties using the concept of an ``almost analytic cocycle construction procedure'' (a.a.c.c.p.). In particular, it is shown that each real valued cocycle may be smoothed (i.e., may be replaced by a cohomologous real analytic cocycle), and in fact the construction in the earlier paper, on a perhaps smaller residual set of \(\alpha\)'s, is an a.a.c.c.p. In addition, by modifying the methods of \textit{A. Iwanik} and \textit{J. Serafin} [Colloq. Math. 66, No. 1, 63-76 (1993)], the construction may be done to give examples of Anzai skew products which are rank one and have partially continuous spectrum.
scientific article; zbMATH DE number 645164